How many poker hands of five cards are dealt from a deck of 52 cards? Also, how many ways are there to select 47 cards from a standard deck of 52 cards ?

To solve these questions, we need to calculate combinations since we're selecting a certain number of cards from a deck without regard to the order.

### 1. **How many poker hands of five cards are dealt from a deck of 52 cards?**

This is equivalent to finding the number of ways to choose 5 cards from 52. The formula for combinations is:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Where:
- \( n = 52 \) (the total number of cards),
- \( k = 5 \) (the number of cards to be selected).

\[
\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5! \cdot 47!}
\]

Calculating this gives:

\[
\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960
\]

So, there are **2,598,960** possible poker hands of five cards dealt from a deck of 52 cards.

---

### 2. **How many ways are there to select 47 cards from a standard deck of 52 cards?**

In this case, we're choosing 47 cards from 52, which is similar to choosing 5 cards from 52, since selecting 47 cards leaves 5 unselected (the unselected ones form a 5-card subset). Thus, the number of ways to select 47 cards is the same as the number of ways to select 5 cards:

\[
\binom{52}{47} = \binom{52}{5} = 2,598,960
\]

Thus, there are **2,598,960** ways to select 47 cards from a deck of 52 cards.

---

### Summary:
- **Number of poker hands of five cards:** 2,598,960
- **Number of ways to select 47 cards from 52 cards:** 2,598,960

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:
1. How many ways can you select 4 cards from a deck of 52?
2. What is the probability of being dealt a royal flush in poker?
3. How many different ways can you select 13 cards from a deck of 52?
4. What is the probability of getting a full house in a 5-card poker hand?
5. How many ways can you choose 5 cards if the deck includes jokers?

**Tip:** When choosing \( k \) items from \( n \), remember that \( \binom{n}{k} = \binom{n}{n-k} \). This symmetry can simplify calculations.

How many poker hands of five cards are dealt from a deck of 52 cards? Also, how many ways are there to select 47 cards from a standard deck of 52 cards?

Find out how many poker hands of five cards can be dealt from a deck of 52 cards using combinatorics. This problem explores key concepts like combinations and symmetry, with a focus on calculating ways to select cards. Perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation